A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

Abstract

We state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution an entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub- and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (comparison theorem). We next study a Bhatnagar--Gross--Krook-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the comparison theorem.